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I have data from the Dutch Cancer Registry in which they calculated the lifetime risk of colon cancer and rectal cancer separately. I wish to use the data for colorectal cancer as a whole. Can I just add up these numbers? If not, what is the appropriate approach?

Ceailloig
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Call the event of getting colon cancer $C$, and the event of getting colorectal cancer $R$. Then

$$P(C \text{ or } R) = P(C) + P(R) - P(C\text{ and } R)$$

by this rule. Then you've got three options:

  • Find out what $P(C \text{ and } R)$ is.
  • Assume $C$ and $R$ are independent, so $P(C \text{ and } R) = P(C)P(R)$.
  • Use a bound rather than exact probability: $0 \leq P(C \text{ and } R) \leq \min \{ P(C), P(R) \}. $
Andy Jones
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Since it is unlikely that there were any patients with both colon cancer and rectal cancer (strictly speaking), the risk can simply be added up.

rnso
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    Why do you think it's unlikely? There's some reasons to imagine it might happen [reasonably often](https://en.wikipedia.org/wiki/Metastasis)? – Glen_b Apr 06 '15 at 08:44
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    The OP has to decide if the two can occur together or not. Metastases are secondary tumors, the primary tumor would be from either colonic or rectal and would be classified as such. – rnso Apr 06 '15 at 08:50
  • It's not just metastasis (I mentioned it because I thought you'd have heard of it and it might lead you to consider whether multiple cancer might ve possible). It seems that [multiple primary cancers are not all that rare](http://www.ncbi.nlm.nih.gov/pubmed/19109776). – Glen_b Apr 06 '15 at 09:04
  • It is more of a medical rather than statistical topic. The fact there is a relatively recent article on this topic means it is not that common occurrence. This article is for cancer survivors, that too long term survivors more than age of 60 years. The large dutch registry does not even have an option to search for dual primary cancers: http://www.cijfersoverkanker.nl/selecties/dataset_1/img5522875c2b390 Also see: http://www.hindawi.com/journals/cripu/2014/760631/ and http://www.oncolink.org/experts/article.cfm?id=1969 – rnso Apr 06 '15 at 13:22
  • (contd). Nonetheless, I agree that OP should also keep a possibility of dual primary cancers, even though its incidence is currently small. – rnso Apr 06 '15 at 13:30